{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T23:58:49Z","timestamp":1762300729804,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T00:00:00Z","timestamp":1718064000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62175032","2023G007","2024CXY106"],"award-info":[{"award-number":["62175032","2023G007","2024CXY106"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Technology Innovation Research and Industrialization Key Projects of Fujian Province","award":["62175032","2023G007","2024CXY106"],"award-info":[{"award-number":["62175032","2023G007","2024CXY106"]}]},{"name":"Fujian Science &amp; Technology Innovation Laboratory for Optoelectronic Information of China","award":["62175032","2023G007","2024CXY106"],"award-info":[{"award-number":["62175032","2023G007","2024CXY106"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>To robustly and adaptively reconstruct displacement, we propose the amplitude modulation integral reconstruction method (AM-IRM) for displacement sensing in a self-mixing interferometry (SMI) system. By algebraically multiplying the SMI signal with a high-frequency sinusoidal carrier, the frequency spectrum of the signal is shifted to that of the carrier. This operation overcomes the issue of frequency blurring in low-frequency signals associated with continuous wavelet transform (CWT), enabling the precise extraction of the Doppler frequency of the SMI signal. Furthermore, the synchrosqueezing wavelet transform (SSWT) is utilized to enhance the frequency resolution of the Doppler signal. Our experimental results demonstrate that the proposed method achieves a displacement reconstruction accuracy of 21.1 nm (0.89%). Additionally, our simulations demonstrated that this method can accurately reconstruct target displacement under the conditions of time-varying optical feedback intensity or a signal-to-noise ratio (SNR) of 0 dB, with a maximum root mean square (RMS) error of 22.2 nm. These results highlight its applicability in real-world environments. This method eliminates the need to manually determine the window length for time\u2013frequency conversion, calculate the parameters of the SMI system, or add additional optical devices, making it easy to implement.<\/jats:p>","DOI":"10.3390\/s24123785","type":"journal-article","created":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T12:11:00Z","timestamp":1718107860000},"page":"3785","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Displacement Sensing for Laser Self-Mixing Interferometry by Amplitude Modulation and Integral Reconstruction"],"prefix":"10.3390","volume":"24","author":[{"given":"Yidan","family":"Huang","sequence":"first","affiliation":[{"name":"National & Local United Engineering Laboratory of Flat Panel Display Technology, College of Physics and Information Engineering, Fuzhou University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenzong","family":"Lai","sequence":"additional","affiliation":[{"name":"National & Local United Engineering Laboratory of Flat Panel Display Technology, College of Physics and Information Engineering, Fuzhou University, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9808-7252","authenticated-orcid":false,"given":"Enguo","family":"Chen","sequence":"additional","affiliation":[{"name":"National & Local United Engineering Laboratory of Flat Panel Display Technology, College of Physics and Information Engineering, Fuzhou University, Fuzhou 350108, China"},{"name":"Fujian Science & Technology Innovation Laboratory for Optoelectronic Information of China, Fuzhou 350108, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"S283","DOI":"10.1088\/1464-4258\/4\/6\/371","article-title":"Laser diode self-mixing technique for sensing applications","volume":"4","author":"Giuliani","year":"2002","journal-title":"J. Opt. A Pure Appl. Opt."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3577","DOI":"10.1109\/JLT.2016.2583919","article-title":"Single-Arm Self-Mixing Superluminescent Diode Interferometer for Flow Measurements","volume":"35","author":"Cattini","year":"2017","journal-title":"J. Light. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"034104","DOI":"10.1117\/1.OE.54.3.034104","article-title":"Laser diode self-mixing interferometry for velocity measurements","volume":"54","author":"Alexandrova","year":"2015","journal-title":"Opt. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4863","DOI":"10.1109\/JLT.2022.3170459","article-title":"All-Fiber Laser-Self-Mixing Interferometer with Adjustable Injection Intensity for Remote Sensing of 40 km","volume":"40","author":"Li","year":"2022","journal-title":"J. Light. Technol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.1109\/TIM.2006.876544","article-title":"Displacement measurements using a self-mixing laser diode under moderate feedback","volume":"55","author":"Bes","year":"2006","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1364\/OPTICA.4.000729","article-title":"Single-spot two-dimensional displacement measurement based on self-mixing interferometry","volume":"4","author":"Zhu","year":"2017","journal-title":"Optica"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4062","DOI":"10.1109\/JLT.2020.3021703","article-title":"All-Fiber Laser-Self-Mixing Sensor for Acoustic Emission Measurement","volume":"39","author":"Liu","year":"2021","journal-title":"J. Light. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIM.2023.3251409","article-title":"Augmentation-Assisted Robust Fringe Detection on Unseen Experimental Signals Applied to Optical Feedback Interferometry Using a Deep Network","volume":"72","author":"Khurshid","year":"2023","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"021201","DOI":"10.3788\/COL201614.021201","article-title":"Simultaneous measurement of vibration amplitude and rotation angle based on a single-channel laser self-mixing interferometer","volume":"14","author":"Sun","year":"2016","journal-title":"Chin. Opt. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"105866","DOI":"10.1016\/j.optlaseng.2019.105866","article-title":"An improved intersection feedback micro-radian angle-measurement system based on the Laser self-mixing interferometry","volume":"126","author":"Zhao","year":"2020","journal-title":"Opt. Lasers Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1844","DOI":"10.1364\/OL.35.001844","article-title":"Thickness measurement of transparent plates by a self-mixing interferometer","volume":"35","author":"Fathi","year":"2010","journal-title":"Opt. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.optcom.2016.05.087","article-title":"New method for lens thickness measurement by the frequency-shifted confocal feedback","volume":"380","author":"Tan","year":"2016","journal-title":"Opt. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"100002","DOI":"10.1063\/1.4952694","article-title":"Self-Mixing Laser Sensor for Short-Distances Measurement","volume":"1740","author":"Norgia","year":"2016","journal-title":"AIP Conf. Proc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.optlaseng.2016.10.013","article-title":"Self-mixing instrument for simultaneous distance and speed measurement","volume":"99","author":"Norgia","year":"2017","journal-title":"Opt. Laser Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4217","DOI":"10.1109\/JLT.2020.3043331","article-title":"An All-Fiber Self-Mixing Range Finder with Tunable Fiber Ring Cavity Laser Source","volume":"39","author":"Zhao","year":"2021","journal-title":"J. Light. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Cavedo, F., Esmaili, P., and Norgia, M. (2022). Self-Mixing Laser Distance-Sensor Enhanced by Multiple Modulation Waveforms. Sensors, 22.","DOI":"10.3390\/s22218456"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1117\/1.1528949","article-title":"Self-mixing in a diode laser as a method for cardiovascular diagnostics","volume":"8","author":"Meigas","year":"2003","journal-title":"J. Biomed. Opt."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.optlastec.2014.04.004","article-title":"A new method for the acquisition of arterial pulse wave using self-mixing interferometry","volume":"63","author":"Arasanz","year":"2014","journal-title":"Opt. Laser Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"990","DOI":"10.1109\/LPT.2004.824631","article-title":"Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical, feedback self-mixing effect","volume":"16","author":"Yu","year":"2004","journal-title":"IEEE Photonics Technol. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"125203","DOI":"10.1016\/j.optcom.2019.125203","article-title":"Effect of linewidth enhancement factor on displacement reconstruction and immediate estimation of feedback factor for weak feedback","volume":"461","author":"Kim","year":"2020","journal-title":"Opt. Commun."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"10587","DOI":"10.1109\/JSEN.2021.3060740","article-title":"Sparsity Promoting Frequency Sampling Method for Estimation of Displacement and Self-Mixing Interferometry Parameters","volume":"21","author":"Khan","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"127751","DOI":"10.1016\/j.optcom.2021.127751","article-title":"Fast and highly accurate estimation of feedback coupling factor and linewidth enhancement factor for displacement sensing under different feedback regimes","volume":"508","author":"Orakzai","year":"2022","journal-title":"Opt. Commun."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1109\/3.341714","article-title":"Laser-Diode Feedback Interferometer for Measurement of Displacements without Ambiguity","volume":"31","author":"Donati","year":"1995","journal-title":"IEEE J. Quantum Electron."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6500207","DOI":"10.1109\/JPHOT.2013.2264277","article-title":"Obtaining High Fringe Precision in Self-Mixing Interference Using a Simple External Reflecting Mirror","volume":"5","author":"Wang","year":"2013","journal-title":"IEEE Photonics J."},{"key":"ref_25","first-page":"1","article-title":"Obtaining Scalable Fringe Precision in Self-Mixing Interference Using an Even-Power Fast Algorithm","volume":"9","author":"Wei","year":"2017","journal-title":"IEEE Photonics J."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Zabit, U., Bernal, O.D., and Bosch, T. (2012, January 28\u201331). Time-Frequency Signal Processing for a Self-Mixing Laser Sensor for Vibration Measurement. Proceedings of the 2012 IEEE Sensors, Taipei, Taiwan.","DOI":"10.1109\/ICSENS.2012.6411122"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4962","DOI":"10.1109\/JSEN.2013.2276106","article-title":"Study of Laser Feedback Phase Under Self-Mixing Leading to Improved Phase Unwrapping for Vibration Sensing","volume":"13","author":"Bernal","year":"2013","journal-title":"IEEE Sens. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"11151","DOI":"10.1109\/JSEN.2019.2935087","article-title":"Spectral Processing of Self-Mixing Interferometric Signal Phase for Improved Vibration Sensing Under Weak- and Moderate-Feedback Regime","volume":"19","author":"Zabit","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1364\/OPEX.13.001537","article-title":"Self-mixing interferometer based on sinusoidal phase modulating technique","volume":"13","author":"Guo","year":"2005","journal-title":"Opt. Express"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"127202","DOI":"10.1016\/j.optcom.2021.127202","article-title":"Quadrature phase detection based on phase shift of self-mixing interferometer for real-time displacement measurement","volume":"498","author":"Dong","year":"2021","journal-title":"Opt. Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"111888","DOI":"10.1016\/j.measurement.2022.111888","article-title":"Quadrature phase detection based on a laser self-mixing interferometer with a wedge for displacement measurement","volume":"202","author":"Li","year":"2022","journal-title":"Measurement"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"18062","DOI":"10.1109\/JSEN.2023.3293856","article-title":"Quadrature-Phase Detection of Laser SMI Based on Square-Wave Modulation for Displacement Measurement","volume":"23","author":"Chen","year":"2023","journal-title":"IEEE Sens. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.optlastec.2016.08.013","article-title":"Carrier-separating demodulation of phase shifting self-mixing interferometry","volume":"89","author":"Tao","year":"2017","journal-title":"Opt. Laser Technol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"7229","DOI":"10.1109\/JLT.2022.3201098","article-title":"Robust Signal Extraction Based on Time-Frequency Joint Analysis and GRNN for a Laser SMI System","volume":"40","author":"Ge","year":"2022","journal-title":"J. Light. Technol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"107531","DOI":"10.1016\/j.optlaseng.2023.107531","article-title":"Signal extraction method based on spectral processing for a dual-channel SMI vibration sensor","volume":"164","author":"Ge","year":"2023","journal-title":"Opt. Laser Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.inffus.2016.03.003","article-title":"A review of remote sensing image fusion methods","volume":"32","author":"Ghassemian","year":"2016","journal-title":"Inf. Fusion"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.optcom.2017.01.037","article-title":"Speckle affected fringe detection based on three envelope extraction for self-mixing displacement measurement","volume":"392","author":"Huang","year":"2017","journal-title":"Opt. Commun."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1109\/JSTQE.2014.2381494","article-title":"Robust Method of Stabilization of Optical Feedback Regime by Using Adaptive Optics for a Self-Mixing Micro-Interferometer Laser Displacement Sensor","volume":"21","author":"Bernal","year":"2015","journal-title":"IEEE J. Sel. Top. Quantum Electron."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/12\/3785\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:56:53Z","timestamp":1760108213000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/12\/3785"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,11]]},"references-count":38,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["s24123785"],"URL":"https:\/\/doi.org\/10.3390\/s24123785","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2024,6,11]]}}}